﻿ DGTD用于RCS计算的初步研究
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 雷达学报  2015, Vol. 4 Issue (3): 361–366  DOI: 10.12000/JR15052 0

### 引用本文 [复制中英文]

[复制中文]
Yang Qian, Wei Bing, Li Lin-qian, et al.. Preliminary research on RCS using DGTD[J].Journal of Radars, 2015, 4(3): 361–366.: 10.12000/JR15052.
[复制英文]

### 文章历史

DGTD用于RCS计算的初步研究
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Preliminary Research on RCS Using DGTD
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School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China
Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China
Abstract: Discontinuous Galerkin Time Domain (DGTD) method appears to be very promising which combines the advantages of unstructured mesh in Finite Element Time Domain (FETD) and explicit scheme in Finite Difference Time Domain (FDTD). This paper first describes principle of DGTD base on vector basis function. Secondly, Specific method for incident plane wave is given for scattering problem. At last, the monostatic Radar Cross Section (RCS) of PEC sphere, medium sphere and the PEC bullet are computed by DGTD method. The numerical results illustrate the feasibility and correctness of the presented scheme. The study of this paper is a foundation for analyzing the RCS of complex target.
Key words: Discontinuous Galerkin Time Domain (DGTD)    Finite Difference Time Domain (FDTD)    Finite Element Method (FEM)    Radar Cross Section (RCS)
1 引言

2 基于棱边基函数的3维DGTD

DGTD可以看作由时域有限元方法(Finite Element Time Domain,FETD)改变边界条件的处理方式得到。FETD是从支配方程和边界条件出发将计算区域划分为多个单元后导出矩阵方程并求解的方法，J. M. Jin,J. F. Lee等人在FETD方面做了很多工作[5, 6, 7]。一般来说，FETD方程推导有两种途径，即变分法与Galerkin法。变分法寻找一个适当泛函对应于支配方程和边界条件，Galerkin法通过加权余量来寻找支配方程和边界条件的相应弱解形式。两种方法都会得到一个大型矩阵方程。在FETD中的若干基本概念如基函数、单元积分、局域、全域、Galerkin加权等均可在DGTD中继承使用，而两种有限元算法的局域边界处理方式有差异。

 $\left. \begin{array}{l} \varepsilon \frac{{\partial E}}{{\partial t}} - \nabla \times H + \sigma E = - J\\ \mu \frac{{\partial E}}{{\partial t}} + \nabla \times E + {\sigma _m}H = - {J_m} \end{array} \right\}$ (1)

 图 1 四面体单元示意图 Fig.1 Schematic diagram of tetrahedron

 $\left. \begin{array}{l} \int {v \cdot } \left[ {\varepsilon \frac{{\partial E}}{{\partial t}} - \nabla \times H + \sigma E + J} \right]d\Omega = 0\\ \int {v \cdot } \left[ {\mu \frac{{\partial E}}{{\partial t}} + \nabla \times E + {\sigma _m}H + {J_m}} \right]d\Omega = 0 \end{array} \right\}$ (2)

 $\left. \begin{array}{l} \int {v \! \cdot \! } \left( {\nabla \!\! \times \!\! E} \right)d\Omega \!\! = \!\! \int {\nabla \cdot } \left( {E \!\! \times \!\! v} \right)d\Omega \!\! + \!\! \int {E \cdot } \left( {\nabla \!\! \times \!\! v} \right)d\Omega \!\! = \!\! \int {\mathop n\limits^ \wedge \cdot } \left( {E \!\! \times \!\! v} \right)ds \!\! + \!\! \int {E \cdot } \left( {\nabla \!\! \times \!\! v} \right)d\Omega \\ \!\! \!\! \quad \quad \qquad \qquad \,\,\,\, = \int {v \! \cdot \! } \left( {\mathop n\limits^ \wedge \!\! \times \!\! E} \right)ds \!\! + \!\! \int {E \! \cdot \! } \left( {\nabla \!\! \times \!\! v} \right)d\Omega \\ \!\! \!\! \!\! \!\! \quad \,\,\, \int {v \! \cdot \! } \left( {\nabla \!\! \times \!\! H} \right)d\Omega \!\! = \!\! \int {\nabla \cdot } \left( {H \!\! \times \!\! v} \right)d\Omega \!\! + \!\! \int {H \cdot } \left( {\nabla \!\! \times \!\! v} \right)d\Omega \!\! = \!\! \int {\mathop n\limits^ \wedge \cdot } \left( {H \!\! \times \!\! v} \right)ds \!\! + \!\! \int {H \cdot } \left( {\nabla \!\! \times \!\! v} \right)d\Omega \\ \!\! \!\! \quad \quad \qquad \qquad \,\,\,\, = \!\! \int {v \! \cdot \! } \left( {\mathop n\limits^ \wedge \!\! \times \!\!H} \right)ds \!\! + \!\! \int {H \! \cdot \! } \left( {\nabla \!\! \times \!\! v} \right)d\Omega \!\! \!\! \!\! \!\! \end{array} \right\}$ (3)

 $\left. \begin{array}{l} \mathop n\limits^ \wedge \times {E^ * } = \mathop n\limits^ \wedge \times E + K_E^e\left[ {\mathop n\limits^ \wedge \times \left( {{E^ + } - E} \right)} \right]\\ \quad \quad \qquad \,\,\,+ v_H^e\left[ {\mathop n\limits^ \wedge \times \left( {\mathop n\limits^ \wedge \times \left( {{H^ + } - H} \right)} \right)} \right]\\ \mathop n\limits^ \wedge \times {H^ * } = \mathop n\limits^ \wedge \times H + K_H^e\left[ {\mathop n\limits^ \wedge \times \left( {{H^ + } - H} \right)} \right]\\ \quad \quad \qquad \,\,\,- v_E^e\left[ {\mathop n\limits^ \wedge \times \left( {\mathop n\limits^ \wedge \times \left( {{E^ + } - E} \right)} \right)} \right] \end{array} \right\}$ (4)

$k_{\rm{H}}^e$,$v_{\rm{H}}^e$,$v_{\rm{E}}^e$的取值如表 1[4]

 $\left. \begin{array}{l} \int {v \! \cdot \! } \left( {\nabla \!\! \times \!\! {E^ * }} \right)d\Omega \!\! = \!\! \int {v \! \cdot \! } \left( {\nabla \!\! \times \!\! E} \right)d\Omega \!\! + \!\! \int {\left\{ {k_E^e\left[ {\mathop n\limits^ \wedge \times \left( {{E^ + } \!\! - \!\! E} \right)} \right] \!\! + \!\! v_H^e\left[ {\mathop n\limits^ \wedge \times \left( {\mathop n\limits^ \wedge \times \left( {{H^ + }\!\! - \!\! H} \right)} \! \right)} \! \right]} \! \right\}} \! ds \!\! \!\! \\ \int {v \! \cdot \! } \left( {\nabla \!\! \times \!\! {H^ * }} \right)d\Omega \!\! = \!\! \int {v \! \cdot \! } \left( {\nabla \!\! \times \!\! H} \right)d\Omega \!\! + \!\! \int {\left\{ {k_H^e\left[ {\mathop n\limits^ \wedge \times \left( {{H^ + } \!\! - \!\! H} \right)} \right] \!\! + \!\! v_E^e\left[ {\mathop n\limits^ \wedge \times \left( {\mathop n\limits^ \wedge \times \left( {{E^ + } \!\! - \!\! E} \right)} \! \right)} \! \right]} \! \right\}} \! ds \!\! \!\! \end{array} \right\}$ (5)

 $\left. \begin{array}{l} \varepsilon \int {v \cdot } \frac{{\partial E}}{{\partial t}}d\Omega \!\! - \!\! \int {v \! \cdot \! } \left( {\nabla \! \times \! H} \right)d\Omega \!\! - \!\! \int {\left\{ {k_H^e\left[ {\mathop n\limits^ \wedge \!\! \times \!\! \left( {{H^ + } \! - \! H} \right)} \right]ds \!\! + \!\! \int {v_E^e} \left[ {\mathop n\limits^ \wedge \!\! \times \!\! \left( {\mathop n\limits^ \wedge \!\! \times \!\! \left( {{E^ + } \! - \! E} \right)} \! \right)} \! \right]} \! \right\}} \! ds \!\!\! \\ \qquad \qquad \,\,\,\,\, + \sigma \int {v \cdot } Ed\Omega + \int {v \cdot } Jd\Omega = \!\! 0 \!\! \\ \mu \int {v \cdot } \frac{{\partial H}}{{\partial t}}d\Omega \!\! + \!\! \int {v \! \cdot \! } \left( {\nabla \! \times \! E} \right)d\Omega \!\! + \!\! \int {\left\{ {k_E^e\left[ {\mathop n\limits^ \wedge \!\! \times \!\! \left( {{E^ + } \! - \! E} \right)} \right]ds \!\! + \!\! \int {v_H^e} \left[ {\mathop n\limits^ \wedge \!\! \times \!\! \left( {\mathop n\limits^ \wedge \!\! \times \!\! \left( {{H^ + } \! - \! H} \right)} \! \right)} \! \right]} \! \right\}} \! ds \!\!\! \\ \qquad \qquad \,\,\,\,\,\, + {\sigma _m}\int {v \cdot } Hd\Omega + \int {v \cdot } {J_m}d\Omega \!\! = 0 \!\! \end{array} \right\}$ (6)

 $\left. \begin{array}{l} \varepsilon \left[ {{M^e}} \right]\frac{\partial }{{\partial t}}\left\{ {{E^e}} \right\} - \left[ {{S^e}} \right]\left\{ {{H^e}} \right\} - \left\{ {F{h^e}} \right\} + \sigma \left[ {{M^e}} \right]\left\{ {{E^e}} \right\} + \left\{ {{J^e}} \right\} = 0\\ \mu \left[ {{M^e}} \right]\frac{\partial }{{\partial t}}\left\{ {{H^e}} \right\} + \left[ {{S^e}} \right]\left\{ {{E^e}} \right\} + \left\{ {F{e^e}} \right\} + {\sigma _m}\left[ {{M^e}} \right]\left\{ {{H^e}} \right\} + \left\{ {{J^e}_m} \right\} = 0 \end{array} \right\}$ (7)

 $\left. {\begin{array}{*{20}{l}} {M_{ij}^e = \iiint\limits_{{\Omega ^e}} {N_i^e \cdot }N_j^ed\Omega ,S_{ij}^e = \iiint\limits_{{\Omega ^e}} {N_i^e \cdot }(\nabla \times N_j^e)d\Omega } \\ {Fh_i^e = \sum {\left[ {k_H^e} \right]} \cdot \left\{ {H_j^{e + } - H_j^e} \right\} - \sum {\left[ {\nu _E^e} \right]} } \\ {\quad \quad \quad \; \cdot \left\{ {E_j^{e + } - E_j^e} \right\}} \\ {\quad \quad = \sum {k_H^e\left[ {Mf_{ij}^e} \right]} \cdot \left\{ {H_j^{e + } - H_j^e} \right\} - \sum {\nu _E^e\left[ {M{\text{g}}_{ij}^e} \right] \cdot \left\{ {E_j^{e + } - E_j^e} \right\}} } \\ {F{\text{e}}_i^e = \sum {\left[ {k_H^e} \right]} \cdot \left\{ {E_j^{e + } - E_j^e} \right\} + \sum {\left[ {\nu _H^e} \right]} } \\ { \quad \quad \quad \; \cdot \left\{ {H_j^{e + } - H_j^e} \right\}} \\ { \quad \quad = \sum {k_E^e\left[ {Mf_{ij}^e} \right]} \cdot \left\{ {E_j^{e + } - E_j^e} \right\} - \sum {\nu _H^e\left[ {M{\text{g}}_{ij}^e} \right] \cdot \left\{ {H_j^{e + } - H_j^e} \right\}} } \\ { \; \; J_i^e = \iiint\limits_{{\Omega ^e}} {N_i^e \cdot } \cdot J_i^ed\Omega ,J_{mi}^e = \iiint\limits_{{\Omega ^e}} {N_i^e \cdot } \cdot J_m^ed\Omega } \end{array}} \right\}$ (8)

3 DGTD中的平面波引入及总场边界条件

 图 2 总场区与散射场区的划分 Fig.2 The total field region and the scattered field region

 图 3 DGTD总场散射场边界 Fig.3 TF/SF boundary for DGTD

(1) 当四面体处于总场边界的总场区一侧时，应在Numerical Flux上加上入射波值，此时$E_j^e$,$H_j^e$属于总场，$E_j^{e+}$,$H_j^{e+}$属于散射场，Ei,Hi为入射波。

 $\left. \begin{array}{1} {\rm{Fh}}_j^e = \sum\limits {k_{\rm{H}}^e\left[{{\rm{Mf}}_{ij}^e} \right] \cdot \left\{ {H_j^{e + } - H_j^e + {\rm{Hi}}} \right\}} \\ \quad\quad\quad - \sum\limits {v_{\rm{E}}^e\left[{{\rm{Mg}}_{ij}^e} \right] \cdot \left\{ {E_j^{e + } - E_j^e + {\rm{Ei}}} \right\}} \\ {\rm{Fe}}_j^e = \sum\limits {k_{\rm{E}}^e\left[{{\rm{Mf}}_{ij}^e} \right] \cdot \left\{ {E_j^{e + } - E_j^e + {\rm{Ei}}} \right\}} \\ \quad\quad\quad + \sum\limits {v_{\rm{H}}^e\left[{{\rm{Mg}}_{ij}^e} \right] \cdot \left\{ {H_j^{e + } - H_j^e + {\rm{Hi}}} \right\}} \end{array} \!\!\! \right\}$ (9)

(2) 当四面体处于总场边界的散射场区一侧时，应在Numerical Flux上减去入射波值，此时3$E_j^e$,$H_j^e$属于散射场，$E_j^{e+}$,$H_j^{e+}$属于总场，Ei,Hi为入射波。

 $\left. \begin{array}{l} {\rm{Fh}}_j^e = \sum\limits {k_{\rm{H}}^e\left[{{\rm{Mf}}_{ij}^e} \right] \cdot \left\{ {H_j^{e + } - H_j^e - {\rm{Hi}}} \right\}} \\ \quad\quad\quad - \sum\limits {v_{\rm{E}}^e\left[{{\rm{Mg}}_{ij}^e} \right] \cdot \left\{ {E_j^{e + } - E_j^e - {\rm{Ei}}} \right\}} \\ {\rm{Fe}}_j^e = \sum\limits {k_{\rm{E}}^e\left[{{\rm{Mf}}_{ij}^e} \right] \cdot \left\{ {E_j^{e + } - E_j^e - {\rm{Ei}}} \right\}} \\ \quad\quad\quad + \sum\limits {v_{\rm{H}}^e\left[{{\rm{Mg}}_{ij}^e} \right] \cdot \left\{ {H_j^{e + } - H_j^e - {\rm{Hi}}} \right\}} \end{array} \!\!\! \right\}$ (10)

4 算例

 图 4 计算域剖分截面图 Fig.4 Sectional view of the computational domain

 图 5 金属球单站RCS Fig.5 The monostatic RCS of PEC sphere

 图 6 介质球单站RCS Fig.6 The monostatic RCS of medium sphere

 图 7 弹头模型 Fig.7 Model of PEC bullet

 图 8 金属弹头单站RCS Fig.8 The monostatic RCS of PEC bullet
5 结论

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